Download App

Brief Review of Cartesian System of Rectangular Coordinates — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSBrief Review of Cartesian System of Rectangular Coordinates4 Marks
Question
A point moves as so that the difference its distances from (ae, 0) and (-ae, 0) is 2 a, prove that the equation to its locus is $\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1 $, where $\text{b}^2=\text{a}^2(\text{e}^2-1).$
✓

Answer

Let P(h, k) be any point on the locus and let A(ae, 0) and B(-ae, 0) be the given points. By the given condition$\text{PA}-\text{PB}=2\text{a}$
$\Rightarrow \text{PA}=2\text{a}+\text{PB}$
$\Rightarrow \sqrt{(\text{ae}-\text{h})^2+(0-\text{k})^2}=2\text{a}+\sqrt{(-\text{ae}-\text{h})^2+(0-\text{k})^2}$
$\Rightarrow (\text{ae}-\text{h})^2+\text{k}^2= \big(2\text{a}+\sqrt{(\text{ae}-\text{h}^2)+\text{k}^2}$ [Taking square on both sides]
$\Rightarrow \text{(ae)}^2+\text{h}^2-2\text{aeh}+\text{k}^2 =4\text{a}^2+(\text{a}\text{e}+\text{h})^2\\\ \ \ +\text{k}^2+2\times2\text{a}\times\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$
$\Rightarrow \text{h}^2+\text{k}^2+\text{(ae)}^2-2\text{aeh} =4\text{a}^2+(\text{a}\text{e})^2+\text{h}^2\\\ \ \ +2\text{h}\text{a}\text{e}+\text{k}^2+4\text{a}\times\sqrt{(\text{a}\text{e}^2+\text{h}^2)+\text{k}^2}$
$\Rightarrow -4\text{a}-2\text{a}\text{e}\text{h}-2\text{a}\text{e}\text{h}=4\text{a}\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$
$\Rightarrow -4\text{a}^2-4\text{a}\text{e}\text{h}=4\text{a}\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$
$\Rightarrow -4\big[\text{a}^2+\text{a}\text{e}\text{h}\big]=4\text{a}\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$
$\Rightarrow -\big[\text{a}^2+\text{a}\text{e}\text{h}\big]=\text{a}\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$
$\Rightarrow -\text{a}[\text{a}+\text{e}\text{h}\big]=\text{a}\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$ $\Rightarrow -[\text{a}+\text{e}\text{h}\big]=\sqrt{(\text{a}\text{e}+\text{h})^2+\text{k}^2}$ $\Rightarrow (\text{a}+\text{e}\text{h})^2=\bigg(\sqrt{(\text{a}\text{e}+\text{h}^2)+\text{k}^2}\bigg)^2$ [Taking square on both sides] $\Rightarrow \text{a}^2+(\text{e}\text{h})^22\text{h}\text{a}\text{e}=(\text{a}\text{e}+\text{h})^2+\text{k}^2$ $\Rightarrow \text{a}^2+(\text{e}\text{h})^2+2\text{h}\text{a}\text{e}=(\text{a}\text{e})+\text{h}^2+2\text{h}\text{a}\text{e}+\text{k}^2$ $\Rightarrow \text{a}^2+\text{e}^2\text{h}^2=\text{a}^2\text{e}^2+\text{h}^2+\text{k}^2$ $\Rightarrow \text{e}^2\text{h}^2-\text{h}^2-\text{k}^2=\text{a}^2\text{e}^2-\text{a}^2$ $\Rightarrow \text{h}^2(\text{e}^2-1)-\text{k}^2=\text{a}^2(\text{e}^2-1)$ $\Rightarrow \frac{\text{h}^2(\text{e}^2-1)}{\text{a}^2(\text{e}^2-1)}-\frac{\text{k}^2}{\text{a}^2(\text{e}^2-1)}=1$ $\Rightarrow \frac{\text{h}^2}{\text{a}^2}-\frac{\text{k}^2}{\text{b}^2},\text{Where }\text{b}^2=\text{a}^2(\text{e}^2-1)$ $\therefore$ The locus of (h, k) is $\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1 $ Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Brief Review of Cartesian System of Rectangular Coordinates→Brief Review of Cartesian System of Rectangular Coordinates — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following from the first principle$\text{f}(\text{x})=\frac{\sin\text{x}}{\text{x}}$
→
Find the angle between the line joining the points (2, 0), (0, 3) and the line x + y = 1.
→
Find the equation to the ellipse in the following case: eccentricity $\text{e}=\frac{1}{2}$ and major axis = 12
→
Find the equation of the parabola whose: Focus is (3, 0) and the directrix is 3 x + 4y = 1
→
If $\sin(\alpha+\beta)=1$ and $\sin(\alpha-\beta)=\frac{1}{2}$ where $0\leq\alpha,\beta\leq\frac{\pi}{2}$ than find the values of $\tan(\alpha+2\beta)$and $\tan(2\alpha+\beta).$
→
From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?
→
At the foot of a mountain, the elevation of it summit is 45°; after ascending 1000m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain.
→
Find the equation of the hyperbola whose,Focus is (2, 2) directrix is $\text{x+y}=\text{9}$ and eccentricity = 2
→
Prove the following identities: $(1+\tan\alpha\tan\beta)^2+(\tan\alpha-\tan\beta)^2=\sec^2\alpha\sec^2\beta$
→
The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observation were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observation were omitted.
→