Download App

Hyperbola — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsHyperbola5 Marks
Question
focus is $(2, 2),$ directive is $x + y = 9$ and eccentricity $= 2.$
✓

Answer

Let $S (2, 2)$ be the focus and $P(x, y)$ be a point on the hyperbola.
Draw PM perpendicular from $P$ on the directrix. Then, by definition
$sP = ePM$
$\Rightarrow sP^2 = e^2PM^2$ 
$\Rightarrow(\text{x}-2)^2+(\text{y}-2)^2=2^2\Big[\frac{\text{x}+\text{y}-9}{\sqrt{1^2+1^2}}\Big]$
$\Big[\because\ \text{e}=\frac{4}{3}\Big]$
$\Rightarrow\ \text{x}^2+4-4\text{x}+\text{y}^2+4-4\text{y}=\frac{4[\text{x+y}-9]^2}{2}$
$\Rightarrow x^2 + y^2 - 4x - 4y + 8 = 2[x + y - 9]^2$
$\Rightarrow x^2 + y^2 - 4x - 4y + 8 = 2[x + y + (-9) + 2 \times x \times y + 2 \times y \times (-9) + 2 \times (-9) \times x]$
$\Rightarrow x^2 + y^2 - 4x - 4y + 8 = 2[x^2 + y^2 + 81 + 2xy - 18y + 18x]$
$\Rightarrow x^2 + y^2 - 4x - 4y + 8 = [2x^2 + 2y^2 + 162 + 4xy - 36y + 36x]$
$\Rightarrow 2x^2 - x^2 + 2y^2 - y^2 + 4xy - 36x + 4x - 36y + 4y + 162 - 8 = 0$
$\Rightarrow x^2 + y^2 - 4xy - 32x - 32y + 154 = 0$
This is the required equation of the hyperbola.

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 "Solve the Following Question.(5 Marks)" from Hyperbola→Hyperbola — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Prove that:
$\tan\text{x}\tan\Big(\frac{\pi}{3}-\text{x}\Big)\tan\Big(\frac{\pi}{3}+\text{x}\Big)=\tan3\text{x}$
→
Prove the following by the principle of mathematical induction:
$1 + 2 + 2^2 + ... + 2^n = 2^{n+1} - 1$ for all $\text{n}\in\text{N}.$
→
Life of bulbs produced by two factories $A$ and $B$ are given below:
Length of life $($in hours$):$ $550-650$ $650-750$ $750-850$ $850-950$ $950-1050$
Factory $A: ($Number of bulbs$)$ $10$ $22$ $52$ $20$ $16$
Factory $B: ($Number of bulbs$)$ $8$ $60$ $24$ $16$ $12$
The bulbs of which factory are more consistent from the point of view of length of life?
→
If a and b are the roots of $\text{x}^2-3\text{x}+\text{p}=0$ and c, d are roots $\text{x}^2-12\text{x}+\text{q}=0,$ where a, b, c, d from a G.P. Prove that (q + p) : (q - p) = 17 : 15.
→
$\begin{aligned}
& \cos A+\operatorname{Cos} B+C \cos C=4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \\
& =2 \cdot \cos \left(\frac{A+B}{2}\right) \cdot \cos \left(\frac{A-B}{2}\right)-\left(1-2 \sin ^2 \frac{C}{2}\right)
\end{aligned}$
→
Find the equations to the straight lines which go through the origin and trisect the portion of the straight line 3x + y = 12 which is intercepted between the axes of coordinates.
→
Show that $\sin100^\circ-\sin10^\circ$ is positive.
→
At What point the origin be shifted so that the equation $x^2+ xy - 3x - y + 2 = 0$ cantain any first degree term and constand term?
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{4}}\frac{2-\text{cosec}^2\text{x}}{1-\cot\text{x}}$
→
Find the number of:
  1. Diagonals.
  2. Triangles formed in a decagon.
→