If p and q are the x and y-intercepts respect | vedclass

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

Question

(C) The equation of the line in intercept form is $\frac{x}{p} + \frac{y}{q} = 1$. Since the line passes through $(a \cos \alpha, b \sin \alpha)$ and

Vedclass · Accepted Answer

$\sec ^2\left(\frac{\alpha-\beta}{2}\right)$

Vedclass · Answer

(C) The equation of the line in intercept form is $\frac{x}{p} + \frac{y}{q} = 1$. Since the line passes through $(a \cos \alpha, b \sin \alpha)$ and $(a \cos \beta, b \sin \beta)$,we have: $\frac{a \cos \alpha}{p} + \frac{b \sin \alpha}{q} = 1$ $(i)$ $\frac{a \cos \beta}{p} + \frac{b \sin \beta}{q} = 1$ (ii) Subtracting (ii) from $(i)$: $\frac{a}{p}(\cos \alpha - \cos \beta) + \frac{b}{q}(\sin \alpha - \sin \beta) = 0$ $\frac{a}{p}(-2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}) = -\frac{b}{q}(2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2})$ $\frac{a}{p} \sin \frac{\alpha+\beta}{2} = \frac{b}{q} \cos \frac{\alpha+\beta}{2} = k$ (let) Then $\frac{a}{p} = k \csc \frac{\alpha+\beta}{2}$ and $\frac{b}{q} = k \sec \frac{\alpha+\beta}{2}$. Substituting into $(i)$: $k \csc \frac{\alpha+\beta}{2} \cos \alpha + k \sec \frac{\alpha+\beta}{2} \sin \alpha = 1$. Solving for $k$ and simplifying the expression $\frac{a^2}{p^2} + \frac{b^2}{q^2}$ leads to $\sec^2 \left(\frac{\alpha-\beta}{2}\right)$.

If $p$ and $q$ are the $x$ and $y$-intercepts respectively of the line passing through the points $(a \cos \alpha, b \sin \alpha)$ and $(a \cos \beta, b \sin \beta)$,then $\frac{a^2}{p^2}+\frac{b^2}{q^2}=$

Similar Questions

Write the equation of the line passing through the points $(1, -1)$ and $(3, 5)$.

The $y$-intercept of the line passing through $A(6, 1)$ and perpendicular to the line $x - 2y = 4$ is:

The minimum area of the triangle formed by the variable line $3 \cos \theta \cdot x + 4 \sin \theta \cdot y = 12$ and the coordinate axes is

If the three points $(3q, 0)$, $(0, 3p)$, and $(1, 1)$ are collinear, then which one is true?

The equation of the straight line passing through $(-a, 0)$ and forming a triangle with the coordinate axes of area $T$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module