If the tangents drawn at the points P and Q o | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The equation of the parabola is $y^{2} = 2x - 3$. The equation of the chord of contact $PQ$ for the point $R(0, 1)$ is given by $T = 0$. $y(1) = 1

Vedclass · Accepted Answer

$(2, -1)$

Vedclass · Answer

(B) The equation of the parabola is $y^{2} = 2x - 3$. The equation of the chord of contact $PQ$ for the point $R(0, 1)$ is given by $T = 0$. $y(1) = 1(x + 0) - 3 \implies y = x - 3$. Substituting $x = y + 3$ into the parabola equation: $y^{2} = 2(y + 3) - 3 = 2y + 6 - 3 = 2y + 3$. $y^{2} - 2y - 3 = 0 \implies (y - 3)(y + 1) = 0$. Thus,$y = 3$ or $y = -1$. For $y = 3$,$x = 6$. For $y = -1$,$x = 2$. So,the points are $P(2, -1)$ and $Q(6, 3)$ (or vice versa). The slope of $PQ$ is $m_{PQ} = \frac{3 - (-1)}{6 - 2} = \frac{4}{4} = 1$. The slope of $PR$ is $m_{PR} = \frac{1 - (-1)}{0 - 2} = \frac{2}{-2} = -1$. Since $m_{PQ} 	imes m_{PR} = 1 	imes (-1) = -1$,the triangle $PQR$ is a right-angled triangle at $P$. In a right-angled triangle,the orthocentre is the vertex where the right angle is formed. Therefore,the orthocentre is $P(2, -1)$.

If the tangents drawn at the points $P$ and $Q$ on the parabola $y^{2} = 2x - 3$ intersect at the point $R(0, 1)$,then the orthocentre of the triangle $PQR$ is.

Similar Questions

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is

What is the equation of the normal to the parabola $y^2 = 4ax$ at the point $(\frac{a}{m^2}, \frac{2a}{m})$?

For the parabola $y^2 = x$,let $PQ$ be a chord such that one endpoint $P$ is $(4, -2)$ and the chord is perpendicular to the axis of the parabola. What is the slope of the normal at $Q$?

The angle between the tangents drawn to the parabola $y^{2}=12x$ from the point $(-3, 2)$ is (in $^{\circ}$)

If $P$ is a point on the parabola $y^2=8x$ and $A$ is the point $(1,0)$,then the locus of the mid-point of the line segment $AP$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module