Show that the path of a moving point such that its distances from… - Vidyadip

Question

Show that the path of a moving point such that its distances from two lines 3x - 2y = 5 and 3x + 2y = 5 are equal is a straight line.

✓

Answer

Let P(h, k) be a moving point such that it is equidistant from the lines 3x - 2y - 5 = 0 and 3x + 2y - 5 = 0, then $\Big|\frac{3\text{h}-2\text{k}-5}{\sqrt{9+4}}\Big|=\Big|\frac{3\text{h}+2\text{k}-5}{\sqrt{9+4}}\Big|$ $|3\text{h}-2\text{k}-5|=|3\text{h}+2\text{k}-5|$ $4\text{k}=0\Rightarrow\text{k}=0$ or $6\text{h}-10=0\Rightarrow3\text{h}=5$ Hence, the locus of (h, k) is y = 0 or 3x = 5, which are straight lines.

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 The Straight Lines→The Straight Lines — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The perpendicular from the origin to the line y = mx + c meets it at the point (-1, 2). Find the values of m and c.

Find the equation of a line which passes through the point (22, -6) and is such that the intercept of x-axis exceeds the intercept of y-axis by 5.

Using distance formula prove that the following points are collinear: A(3, -5, 1), B(-1, 0, 8) and C(7, -10, -6)

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\tan3\text{x}-2\text{x}}{3\text{x}-\sin^2\text{x}}$

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\cos3\text{x}-\cos5\text{x}}{\text{x}^2}$

Show that the product of perpendiculars on the line $\frac{\text{x}}{\text{a}}\cos\theta+\frac{\text{y}}{\text{b}}\sin\theta=1$ from the points $\Big(\sqrt{\text{a}^2-\text{b}^2},0\Big)$ is $\text{b}^2.$

Prove the following by using the principle of mathematical induction for all $n ∈ N: (2n + 7) < (n + 3)^2$.

Find the maximum and minimum values of each of the following trigonometrical expressions: $\sin\text{x}-\cos\text{x}+1$

Find the numberof observation lying between $\overline{\text{X}}-\text{M.D. }$ and $\overline{\text{X}} +\text{ M.D.}$ is the mean deviation from the mean. $38, 70, 48, 34, 63, 42, 55, 44, 53, 47$

Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:$9\text{x}^2+25\text{y}^2=225$